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Item Harnessing Quantum Systems for Sensing, Simulation, and Optimization(2024) Bringewatt, Jacob Allen; Gorshkov, Alexey V; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Quantum information science offers a remarkable promise: by thinking practically about how quantum systems can be put to work to solve computational and information processing tasks, we gain novel insights into the foundations of quantum theory and computer science. Or, conversely, by (re)considering the fundamental physical building blocks of computers and sensors, we enable new technologies, with major impacts for computational and experimental physics. In this dissertation, we explore these ideas through the lens of three different types of quantum hardware, each with a particular application primarily in mind: (1) networks of quantum sensors for measuring global properties of local field(s); (2) analog quantum computers for solving combinatorial optimization problems; and (3) digital quantum computers for simulating lattice (gauge) theories. For the setting of quantum sensor networks, we derive the fundamental performance limits for the sensing task of measuring global properties of local field(s) in a variety of physical settings (qubit sensors, Mach-Zehnder interferometers, quadrature displacements) and present explicit protocols that achieve these limits. In the process, we reveal the geometric structure of the fundamental bounds and the associated algebraic structure of the corresponding protocols. We also find limits on the resources (e.g. entanglement or number of control operations) required by such protocols. For analog quantum computers, we focus on the possible origins of quantum advantage for solving combinatorial optimization problems with an emphasis on investigating the power of adiabatic quantum computation with so-called stoquastic Hamiltonians. Such Hamiltonians do not exhibit a sign problem when classically simulated via quantum Monte Carlo algorithms, suggesting deep connections between the sign problem, the locality of interactions, and the origins of quantum advantage. We explore these connections in detail. Finally, for digital quantum computers, we consider the optimization of two tasks relevant for simulating lattice (gauge) theories. First, we investigate how to map fermionic systems to qubit systems in a hardware-aware manner that consequently enables an improved parallelization of Trotter-based time evolution algorithms on the qubitized Hamiltonian. Second, we investigate how to take advantage of known symmetries in lattice gauge theories to construct more efficient randomized measurement protocols for extracting purities and entanglement entropies from simulated states. We demonstrate how these protocols can be used to detect a phase transition between a trivial and a topologically ordered phase in $Z_2$ lattice gauge theory. Detecting this transition via these randomized methods would not otherwise be possible without relearning all symmetries.Item APPLICATIONS OF ARTIFICIAL NEURAL NETWORKS IN LEARNING QUANTUM SYSTEMS(2023) Pan, Ruizhi; Clark, Charles; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Quantum machine learning is an emerging field that combines techniques in the disciplines of machine learning (ML) and quantum physics. Research in this field takes three broad forms: applications of classical ML techniques to quantum physical systems, quantum computing and algorithms for classical ML problems, and new ideas inspired by the intersection of the two disciplines. We mainly focus on the power of artificial neural networks (NNs) in quantum-state representation and phase classification in this work. In the first part of the dissertation, we study NN quantum states which are used as wave-function ans{\" a}tze in the context of quantum many-body physics. While these states have achieved success in simulating low-lying eigenstates and short-time unitary dynamics of quantum systems and efficiently representing particular states such as those with a stabilizer nature, more rigorous quantitative analysis about their expressibility and complexity is warranted. Here, our analysis of the restricted Boltzmann machine (RBM) state representation of one-dimensional (1D) quantum spin systems provides new insight into their computational complexity. We define a class of long-range-fast-decay (LRFD) RBM states with quantifiable upper bounds on truncation errors and provide numerical evidence for a large class of 1D quantum systems that may be approximated by LRFD RBMs of at most polynomial complexities. These results lead us to conjecture that the ground states of a wide range of quantum systems may be exactly represented by LRFD RBMs or a variant of them, even in cases where other state representations become less efficient. At last, we provide the relations between multiple typical state manifolds. Our work proposes a paradigm for doing complexity analysis for generic long-range RBMs which naturally yields a further classification of this manifold. This paradigm and our characterization of their nonlocal structures may pave the way for understanding the natural measure of complexity for quantum many-body states described by RBMs and are generalizable for higher-dimensional systems and deep neural-network quantum states. In the second part, we use RBMs to investigate, in dimensions $D=1$ and $2$, the many-body excitations of long-range power-law interacting quantum spin models. We develop an energy-shift method to calculate the excited states of such spin models and obtain a high-precision momentum-resolved low-energy spectrum. This enables us to identify the critical exponent where the maximal quasiparticle group velocity transits from finite to divergent in the thermodynamic limit numerically. In $D=1$, the results agree with an analysis using the field theory and semiclassical spin-wave theory. Furthermore, we generalize the RBM method for learning excited states in nonzero-momentum sectors from 1D to 2D systems. At last, we analyze and provide all possible values ($3/2$, $2$ and $3$) of the critical exponent for 1D generic quadratic bosonic and fermionic Hamiltonians with long-range hoppings and pairings which serves for understanding the speed of information propagation in quantum systems. In the third part, we study deep NNs as phase classifiers. We analyze the phase diagram of a 2D topologically nontrivial fermionic model Hamiltonian with pairing terms at first and then demonstrate that deep NNs can learn the band-gap closing conditions only based on wave-function samples of several typical energy eigenstates, thus being able to identify the phase transition point without knowledge of Hamiltonians.Item DEVELOPING MACHINE LEARNING TECHNIQUES FOR NETWORK CONNECTIVITY INFERENCE FROM TIME-SERIES DATA(2022) Banerjee, Amitava; Ott, Edward; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Inference of the connectivity structure of a network from the observed dynamics of the states of its nodes is a key issue in science, with wide-ranging applications such as determination of the synapses in nervous systems, mapping of interactions between genes and proteins in biochemical networks, distinguishing ecological relationships between different species in their habitats etc. In this thesis, we show that certain machine learning models, trained for the forecasting of experimental and synthetic time-series data from complex systems, can automatically learn the causal networks underlying such complex systems. Based on this observation, we develop new machine learning techniques for inference of causal interaction network connectivity structures underlying large, networked, noisy, complex dynamical systems, solely from the time-series of their nodal states. In particular, our approach is to first train a type of machine learning architecture, known as the ‘reservoir computer’, to mimic the measured dynamics of an unknown network. We then use the trained reservoir computer system as an in silico computational model of the unknown network to estimate how small changes in nodal states propagate in time across that network. Since small perturbations of network nodal states are expected to spread along the links of the network, the estimated propagation of nodal state perturbations reveal the connections of the unknown network. Our technique is noninvasive, but is motivated by the widely used invasive network inference method, whereby the temporal propagation of active perturbations applied to the network nodes are observed and employed to infer the network links (e.g., tracing the effects of knocking down multiple genes, one at a time, can be used infer gene regulatory networks). We discuss how we can further apply this methodology to infer causal network structures underlying different time-series datasets and compare the inferred network with the ground truth whenever available. We shall demonstrate three practical applications of this network inference procedure in (1) inference of network link strengths from time-series data of coupled, noisy Lorenz oscillators, (2) inference of time-delayed feedback couplings in opto-electronic oscillator circuit networks designed the laboratory, and, (3) inference of the synaptic network from publicly-available calcium fluorescence time-series data of C. elegans neurons. In all examples, we also explain how experimental factors like noise level, sampling time, and measurement duration systematically affect causal inference from experimental data. The results show that synchronization and strong correlation among the dynamics of different nodal states are, in general, detrimental for causal network inference. Features that break synchrony among the nodal states, e.g., coupling strength, network topology, dynamical noise, and heterogeneity of the parameters of individual nodes, help the network inference. In fact, we show in this thesis that, for parameter regimes where the network nodal states are not synchronized, we can often achieve perfect causal network inference from simulated and experimental time-series data, using machine learning techniques, in a wide variety of physical systems. In cases where effects like observational noise, large sampling time, or small sampling duration hinder such perfect network inference, we show that it is possible to utilize specially-designed surrogate time-series data for assigning statistical confidence to individual inferred network links. Given the general applicability of our machine learning methodology in time-series prediction and network inference, we anticipate that such techniques can be used for better model-building, forecasting, and control of complex systems in nature and in the lab.Item ON THEORETICAL ANALYSES OF QUANTUM SYSTEMS: PHYSICS AND MACHINE LEARNING(2022) Guo, Shangjie; Spielman, Ian B; Taylor, Jacob M; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Engineered quantum systems can help us learn more about fundamental physics topics and quantum technologies with real-world applications. However, building them could involve several challenging tasks, such as designing more noise-resistant quantum components in confined space, manipulating continuously-measured quantum systems without destroying coherence, and extracting information about quantum phenomena using machine learning (ML) tools. In this dissertation, we present three examples from the three aspects of studying the dynamics and characteristics of various quantum systems. First, we examine a circuit quantum acoustodynamic system consisting of a superconducting qubit, an acoustical waveguide, and a transducer that nonlocally couples both. As the sound signals travel $10^5$ times slower than the light and the coupler dimension extends beyond a few phonon emission wavelengths, we can model the system as a non-Markovian giant atom. With an explicit result, we show that a giant atom can exhibit suppressed relaxation within a free space and an effective vacuum coupling emerges between the qubit excitation and a confined acoustical wave packet. Second, we study closed-loop controls for open quantum systems using weakly-monitored Bose-Einstein condensates (BECs) as a platform. We formulate an analytical model to describe the dynamics of backaction-limited weak measurements and temporal-spatially resolved feedback imprinting. Furthermore, we design a backaction-heating-prevention feedback protocol that stabilizes the system in quasi-equilibrium. With these results, we introduce closed-loop control as a powerful instrument for engineering open quantum systems. At last, we establish an automated framework consisting of ML and physics-informed models for solitonic feature identification from experimental BEC image data. We develop classification and object detection algorithms based on convolutional neural networks. Our framework eliminates human inspections and enables studying soliton dynamics from numerous images. Moreover, we publish a labeled dataset of soliton images and an open-source Python package for implementing our framework.Item The complexity of simulating quantum physics: dynamics and equilibrium(2021) Deshpande, Abhinav; Gorshkov, Alexey V; Fefferman, Bill; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Quantum computing is the offspring of quantum mechanics and computer science, two great scientific fields founded in the 20th century. Quantum computing is a relatively young field and is recognized as having the potential to revolutionize science and technology in the coming century. The primary question in this field is essentially to ask which problems are feasible with potential quantum computers and which are not. In this dissertation, we study this question with a physical bent of mind. We apply tools from computer science and mathematical physics to study the complexity of simulating quantum systems. In general, our goal is to identify parameter regimes under which simulating quantum systems is easy (efficiently solvable) or hard (not efficiently solvable). This study leads to an understanding of the features that make certain problems easy or hard to solve. We also get physical insight into the behavior of the system being simulated. In the first part of this dissertation, we study the classical complexity of simulating quantum dynamics. In general, the systems we study transition from being easy to simulate at short times to being harder to simulate at later times. We argue that the transition timescale is a useful measure for various Hamiltonians and is indicative of the physics behind the change in complexity. We illustrate this idea for a specific bosonic system, obtaining a complexity phase diagram that delineates the system into easy or hard for simulation. We also prove that the phase diagram is robust, supporting our statement that the phase diagram is indicative of the underlying physics. In the next part, we study open quantum systems from the point of view of their potential to encode hard computational problems. We study a class of fermionic Hamiltonians subject to Markovian noise described by Lindblad jump operators and illustrate how, sometimes, certain Lindblad operators can induce computational complexity into the problem. Specifically, we show that these operators can implement entangling gates, which can be used for universal quantum computation. We also study a system of bosons with Gaussian initial states subject to photon loss and detected using photon-number-resolving measurements. We show that such systems can remain hard to simulate exactly and retain a relic of the "quantumness" present in the lossless system. Finally, in the last part of this dissertation, we study the complexity of simulating a class of equilibrium states, namely ground states. We give complexity-theoretic evidence to identify two structural properties that can make ground states easier to simulate. These are the existence of a spectral gap and the existence of a classical description of the ground state. Our findings complement and guide efforts in the search for efficient algorithms.Item LOCALITY, SYMMETRY, AND DIGITAL SIMULATION OF QUANTUM SYSTEMS(2021) Tran, Cong Minh; Gorshkov, Alexey V.; Taylor, Jacob M.; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Besides potentially delivering a huge leap in computational power, quantum computers also offer an essential platform for simulating properties of quantum systems. Consequently, various algorithms have been developed for approximating the dynamics of a target system on quantum computers. But generic quantum simulation algorithms—developed to simulate all Hamiltonians—are unlikely to result in optimal simulations of most physically relevant systems; optimal quantum algorithms need to exploit unique properties of target systems. The aim of this dissertation is to study two prominent properties of physical systems, namely locality and symmetry, and subsequently leverage these properties to design efficient quantum simulation algorithms. In the first part of the dissertation, we explore the locality of quantum systems and the fundamental limits on the propagation of information in power-law interacting systems. In particular, we prove upper limits on the speed at which information can propagate in power-law systems. We also demonstrate how such speed limits can be achieved by protocols for transferring quantum information and generating quantum entanglement. We then use these speed limits to constrain the propagation of error and improve the performance of digital quantum simulation. Additionally, we consider the implications of the speed limits on entanglement generation, the dynamics of correlation, the heating time, and the scrambling time in power-law interacting systems. In the second part of the dissertation, we propose a scheme to leverage the symmetry of target systems to suppress error in digital quantum simulation. We first study a phenomenon called destructive error interference, where the errors from different steps of a simulation cancel out each other. We then show that one can induce the destructive error interference by interweaving the simulation with unitary transformations generated by the symmetry of the target system, effectively providing a faster quantum simulation by symmetry protection. We also derive rigorous bounds on the error of a symmetry-protected simulation algorithm and identify conditions for optimal symmetry protection.Item QUANTUM ALGORITHMS FOR DIFFERENTIAL EQUATIONS(2019) Ostrander, Aaron Jacob; Childs, Andrew; Monroe, Chris; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This thesis describes quantum algorithms for Hamiltonian simulation, ordinary differential equations (ODEs), and partial differential equations (PDEs). Product formulas are used to simulate Hamiltonians which can be expressed as a sum of terms which can each be simulated individually. By simulating each of these terms in sequence, the net effect approximately simulates the total Hamiltonian. We find that the error of product formulas can be improved by randomizing over the order in which the Hamiltonian terms are simulated. We prove that this approach is asymptotically better than ordinary product formulas and present numerical comparisons for small numbers of qubits. The ODE algorithm applies to the initial value problem for time-independent first order linear ODEs. We approximate the propagator of the ODE by a truncated Taylor series, and we encode the initial value problem in a large linear system. We solve this linear system with a quantum linear system algorithm (QLSA) whose output we perform a post-selective measurement on. The resulting state encodes the solution to the initial value problem. We prove that our algorithm is asymptotically optimal with respect to several system parameters. The PDE algorithms apply the finite difference method (FDM) to Poisson's equation, the wave equation, and the Klein-Gordon equation. We use high order FDM approximations of the Laplacian operator to develop linear systems for Poisson's equation in cubic volumes under periodic, Neumann, and Dirichlet boundary conditions. Using QLSAs, we output states encoding solutions to Poisson's equation. We prove that our algorithm is exponentially faster with respect to the spatial dimension than analogous classical algorithms. We also consider how high order Laplacian approximations can be used for simulating the wave and Klein-Gordon equations. We consider under what conditions it suffices to use Hamiltonian simulation for time evolution, and we propose an algorithm for these cases that uses QLSAs for state preparation and post-processing.Item Building and Programming a Universal Ion Trap Quantum Computer(2018) Figgatt, Caroline; Monroe, Christopher; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Quantum computing represents an exciting frontier in the realm of information processing; it is a promising technology that may provide future advances in a wide range of fields, from quantum chemistry to optimization problems. This thesis discusses experimental results for several quantum algorithms performed on a programmable quantum computer consisting of a linear chain of five or seven trapped 171Yb+ atomic clock ions with long coherence times and high gate fidelities. We execute modular one- and two-qubit computation gates through Raman transitions driven by a beat note between counter-propagating beams from a pulsed laser. The system's individual addressing capability provides arbitrary single-qubit rotations as well as all possible two-qubit entangling gates, which are implemented using a pulse-segmentation scheme. The quantum computer can be programmed from a high-level interface to execute arbitrary quantum circuits, and comes with a toolbox of many important composite gates and quantum subroutines. We present experimental results for a complete three-qubit Grover quantum search algorithm, a hallmark application of a quantum computer with a well-known speedup over classical searches of an unsorted database, and report better-than-classical performance. The algorithm is performed for all 8 possible single-result oracles and all 28 possible two-result oracles. All quantum solutions are shown to outperform their classical counterparts. Performing parallel operations will be a powerful capability as deeper circuits on larger, more complex quantum computers present new challenges. Here, we perform a pair of 2-qubit gates simultaneously in a single chain of trapped ions. We employ a pre-calculated pulse shaping scheme that modulates the phase and amplitude of the Raman transitions to drive programmable high-fidelity 2-qubit entangling gates in parallel by coupling to the collective modes of motion of the ion chain. Ensuring the operation yields only spin-spin interactions between the desired pairs, with neither residual spin-motion entanglement nor crosstalk spin-spin entanglement, is a nonlinear constraint problem, and pulse solutions are found using optimization techniques. As an application, we demonstrate the quantum full adder using a depth-4 circuit requiring the use of parallel 2-qubit operations.Item Spectral graph theory with applications to quantum adiabatic optimization(2016) Baume, Michael Jarret; Jordan, Stephen P; Childs, Andrew; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this dissertation I draw a connection between quantum adiabatic optimization, spectral graph theory, heat-diffusion, and sub-stochastic processes through the operators that govern these processes and their associated spectra. In particular, we study Hamiltonians which have recently become known as ``stoquastic'' or, equivalently, the generators of sub-stochastic processes. The operators corresponding to these Hamiltonians are of interest in all of the settings mentioned above. I predominantly explore the connection between the spectral gap of an operator, or the difference between the two lowest energies of that operator, and certain equilibrium behavior. In the context of adiabatic optimization, this corresponds to the likelihood of solving the optimization problem of interest. I will provide an instance of an optimization problem that is easy to solve classically, but leaves open the possibility to being difficult adiabatically. Aside from this concrete example, the work in this dissertation is predominantly mathematical and we focus on bounding the spectral gap. Our primary tool for doing this is spectral graph theory, which provides the most natural approach to this task by simply considering Dirichlet eigenvalues of subgraphs of host graphs. I will derive tight bounds for the gap of one-dimensional, hypercube, and general convex subgraphs. The techniques used will also adapt methods recently used by Andrews and Clutterbuck to prove the long-standing ``Fundamental Gap Conjecture''.Item Duality methods in networks, computer science models, and disordered condensed matter systems(2014) Mitchell, Joe Dan; Galitski, Victor M; Physics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this thesis, I explore lattice independent duality and systems to which it can be applied. I first demonstrate classical duality on models in an external field, including the Ising, Potts, and x-y models, showing in particular how this modifies duality to be lattice independent and applicable to networks. I then present a novel application of duality on the boolean satsifiability problem, one of the most important problems in computational complexity, through mapping to a low temperature Ising model. This establishes the equivalence between boolean satisfiability and a problem of enumerating the positive solutions to a Diophantine system of equations. I continue by combining duality with a prominent tool for models on networks, belief propagation, deriving a new message passing procedure, dual belief propagation. In the final part of my thesis, I shift to propose and examine a semiclassical model, the two-component Coulomb glass model, which can explain the giant magnetoresistance peak present in disordered films near a superconductor-insulator transition as the effect of competition between single particle and localized pair transport. I numerically analyze the density of states and transport properties of this model.